Prior to development of high-speed digital computers, the method of characteristics has not been readily available for pneumatic system design. The traditional organization of computations on a free-running grid is not ideal for machine work. In a 1953 paper, Hartree developed the mathematical foundation for method of characteristics computations on a rectangular grid of predetermined dimensions. The present paper develops in detail a rectangular grid scheme for pneumatic line analysis. An approximate method for tapered lines is presented. Line termination computations are discussed for the general case, together with examples developed in connection with recent resonance tube research conducted by the author. Some computed results for constant area and tapered resonance tube simulations are presented.